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Zipfs law describes the empirical size distribution of the components of many systems in natural and social sciences and humanities. We show, by solving a statistical model, that Zipfs law co-occurs with the maximization of the diversity of the component sizes. The law ruling the increase of such diversity with the total dimension of the system is derived and its relation with Heaps law is discussed. As an example, we show that our analytical results compare very well with linguistics datasets.
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This paper investigates the rank distribution, cumulative probability, and probability density of price returns for the stocks traded in the KSE and the KOSDAQ market. This research demonstrates that the rank distribution is consistent approximately with the Zipfs law with exponent $alpha = -1.00$ (KSE) and -1.31 (KOSDAQ), similar that of stock prices traded on the TSE. In addition, the cumulative probability distribution follows a power law with scaling exponent $beta = -1.23$ (KSE) and -1.45 (KOSDAQ). In particular, the evidence displays that the probability density of normalized price returns for two kinds of assets almost has the form of an exponential function, similar to the result in the TSE and the NYSE.
The origin(s) of the ubiquity of Zipfs law (an inverse power law form for the probability density function (PDF) with exponent $1+1$) is still a matter of fascination and investigation in many scientific fields from linguistic, social, economic, computer sciences to essentially all natural sciences. In parallel, self-excited dynamics is a prevalent characteristic of many systems, from seismicity, financial volatility and financial defaults, to sociology, consumer behaviors, computer sciences, The Internet, neuronal discharges and spike trains in biological neuron networks, gene expression and even criminology. Motivated by financial and seismic modelling, we bring the two threads together by introducing a general class of nonlinear self-excited point processes with fast-accelerating intensities as a function of tension. Solving the corresponding master equations, we find that a wide class of such nonlinear Hawkes processes have the PDF of their intensities described by Zipfs law on the condition that (i) the intensity is a fast-accelerating function of tension and (ii) the distribution of the point fertilities is symmetric. This unearths a novel mechanism for Zipfs law, providing a new understanding of its ubiquity.
A major problem for evolutionary theory is understanding the so called {em open-ended} nature of evolutionary change, from its definition to its origins. Open-ended evolution (OEE) refers to the unbounded increase in complexity that seems to characterise evolution on multiple scales. This property seems to be a characteristic feature of biological and technological evolution and is strongly tied to the generative potential associated with combinatorics, which allows the system to grow and expand their available state spaces. Interestingly, many complex systems presumably displaying OEE, from language to proteins, share a common statistical property: the presence of Zipfs law. Given an inventory of basic items (such as words or protein domains) required to build more complex structures (sentences or proteins) Zipfs law tells us that most of these elements are rare whereas a few of them are extremely common. Using Algorithmic Information Theory, in this paper we provide a fundamental definition for open-endedness, which can be understood as {em postulates}. Its statistical counterpart, based on standard Shannon Information theory, has the structure of a variational problem which is shown to lead to Zipfs law as the expected consequence of an evolutionary process displaying OEE. We further explore the problem of information conservation through an OEE process and we conclude that statistical information (standard Shannon information) is not conserved, resulting into the paradoxical situation in which the increase of information content has the effect of erasing itself. We prove that this paradox is solved if we consider non-statistical forms of information. This last result implies that standard information theory may not be a suitable theoretical framework to explore the persistence and increase of the information content in OEE systems.
We statistically investigate the distribution of share price and the distributions of three common financial indicators using data from approximately 8,000 companies publicly listed worldwide for the period 2004-2013. We find that the distribution of share price follows Zipfs law; that is, it can be approximated by a power law distribution with exponent equal to 1. An examination of the distributions of dividends per share, cash flow per share, and book value per share - three financial indicators that can be assumed to influence corporate value (i.e. share price) - shows that these distributions can also be approximated by a power law distribution with power-law exponent equal to 1. We estimate a panel regression model in which share price is the dependent variable and the three financial indicators are explanatory variables. The two-way fixed effects model that was selected as the best model has quite high power for explaining the actual data. From these results, we can surmise that the reason why share price follows Zipfs law is that corporate value, i.e. company fundamentals, follows Zipfs law.
We propose hypotheses describing the empirical finding of an association between the exponents of urban GDP scaling and Zipfs law for cities. These hypotheses represent various combinations of directional or reciprocal causal links between the two phenomena and include inter- and intra-city processes. Future theories and models can be motivated with and categorized according to these hypotheses. This paper intends to stimulate the discussion around the processes behind these phenomena and pave the way to a Unified Urban Theory.
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